Scientific research activities at the Institute are conducted in accordance with the following four research programmes:

• Nonlinear dynamics and global analysis;
• Principles of mathematics and combinatorics;
• Mechanics of Continua;
• Functional-differential and ordinary differential equations.

Programmes are developed considering existing research potential specificity and maximum coordination. The approximate duration of programmes is 10-15 years.

Main areas of research:

• Nonlinear dynamics and global analysis;
• Geometry and topology of mechanisms and nanostructures;
• Size theory and descriptive set theory;
• Combinatorial and discrete geometry;
• Mathematical models of the theory of elasticity;
• Boundary Value Problems for functional-differential equations.

Main areas of research at the Institute are based on the following four principles:

• Active researches in this field are being conducted in many of the world’s leading mathematical centers;
• Members of the department have sufficient research qualifications, experience and significant results;
• All the aforementioned areas are based on fundamental mathematical theories, yet are promising in terms of practical application;
• Members of the department have constant contact and cooperation with leading international experts in these fields.
• Study in the field of Nonlinear Dynamics and global analysis will include study methods of non-linear systems of mathematical modeling and stability, analytical and qualitative research methods of mathematical physics nonlinear integrated models, algorithmic calculation methods of non-linear system real number solutions to equations, algebraic methods of calculating topological invariants of non-linear plural system solutions, non-linear dynamic systems sustainability and bifurcation issues, algorithmic methods for checking the stability of complex systems, the study of nodes and entangled quantum systems dynamics.
• Study field of Geometry and topology of mechanisms and nanostructures will study the following: configuration spaces mechanisms and robots, extreme tasks of configuration spaces, kinematic features of mechanisms, configuration spaces of tensegrity type systems and nanostructures, sustainable configurations geometry of tensegrity type systems, connections among nanostructure geometry, topography and physical qualities.
• Size theory and descriptive set theory will study further exploration and generalization of notions of measurability of sets and functions (Universal measurability, relative measurability and absolute non-measurability); classification of functions and sets based on these concepts; connections of notions of measurability with features of topography and other issues; Descriptive exotic structure functions and point-sets in terms of size and Barry category.
• Combinatorial and discrete geometry will study: a convex polyhedron combinatorial structure in Euclidean space, the establishment of symmetry criteria for Polyhedron and for more general geometric shapes, main features of discrete point systems and their use in practical problems.
• Mathematical models of the theory of elasticity will study: a matter of three-dimensional equation statics, sustainable fluctuation and dynamics for microstructural solids (mixtures, composites, porous materials, etc.) and resonant processes in these solids, within theories of elasticity, thermoelasticity and micropolar; the existence of individual frequencies of sustainable fluctuation inner boundary equations; key features of flat waves.
• Boundary Value Problems for functional-differential equations examines: boundary theory of linear and nonlinear ordinary differential and functional-differential equations (Cauchy, Cauchy-Nicoletti, Dirichlet, Neumann, periodicals and general boundary value problems), as well as sustainability and correctness issues, asymptotic theory and connection between them, more specifically – nonlinear equations are studied both in cases of resonance and non-resonance, regular and singular equations will also be studied.